An exact integral equation for steady surface waves

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AN EXACT INTEGRAL EQUATION FOR STEADY SURFACE WAVES

John Byatt-Smith

REPORT NO. AS-69-3

CONTRACT NO. Nonr-222(79) FEBRUARY 1969

COLLEGE OF ENGINEERING

AN EXACT INTEGRAL EQUATION FOR STEADY SURFACE WAVES

John Byatt-Smith

This document has been approved for public release and sale; its distribution is unlimited

FACULTY. INVESTIGATOR:

M. HOLT, Professor of Aeronautical Sciences

DIVISION OF AERONAUTICAL SCIENCES UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720

REPORT NO. AS-69-3 U. S. OFFICE OF NAVAL RESEARCH,

ABSTRACT

An exact integral equation is found for inviscid steady surface waves. Existing known approximations are all derived from the one

equa-tion. A numerical solution is attempted in the case of the solitary wave. From the numerical solution the amplitude of the solitary wave of maximum height is estimated. The numerical solution is also compared with the existing approximations and the implications drawn from the comparison are discussed.

INTRODUCTION

The interest in the motion of gravity waves was first stimulated by the experimental studies of Russell (1844). He was the first to report the physical observation of the phenomenon corresponding to the steady finite amplitude solitary wave. At the time, most hydrodynamicists did not believe it could be a permanent wave form. Subsequently Boussinesq in 1871, Rayleigh in 1876 and Korteweg and De Vries in 1885 brought forward conclusive mathematical arguments for its existance. Korteweg and De Vries also went on to show that permanent finite amplitude waves were possible and termed them cnoidal waves.

Following this early work, many attempts were made to study large amplitude solitary waves. In 1894 McCowan used an approximate theory which satisfied the constant pressure condition only at the crest and at infinity. From his numerical solution he predicted that the amplitude of the solitary wave of maximum height was .78 of the free water height.

Packham (1952) in his attempt to get a solution also replaced the constant pressure condition. With his new boundary conditions, he was able to get an exact solution. Packham's method introduced an unknown parameter with his boundary condition. He gave reasons for choosing a certain value that gave the maximum amplitude of 1.03 times the free water depth. How-ever, when the parameter is chosen so as to satisfy the pressure condition at infinity his method of solution gives a maximum amplitude of .83. This figure is in agreement with that obtained by )'amada (1957) and Lenau (1965). They assume that the complex velocity potential could be expressed in

2

coefficients. _{A numerical solution was then obtained, by truncating} this power series. All the above investigations assumed the Stokes

criterion that the solitary wave of maximum amplitude has a 120° corner

at the crest and sought for a solution of this form.

While the solitary wave of maximum height has attracted the attention of many authors, the attention given to solitary waves and cnoidal waves of large amplitude has been sparse. It was not until 1948 that Friedrichs introduced a systematic perturbation solution to improve the solution obtained by Korteweg and De Vries. Keller (1948) carried out the necessary series expansion to show that this formal procedure did yield the known first approximations to the finite amplitude, shallow water waves. In 1962 Laitone extended the expansion to obtain the second order solution to shallow water waves.

In 1955 Long used a different approach to the problem of large amplitude solitary waves. Instead of finding the complex velocity

potential in terms of x and. y , the horizontal and vertical independent

variables, he inverted the problem and found, x and y in 'ternis of the

velocity potential and the stream function. He did not obtain an ex-plicit solution but left it in the form of a first order differential equation that could be solved numerically. Although Long confined his attention to the solitary wave, it is possible to use his method to obtain cnoidal waves.

In 1921 Nekrasov transformed the problem of steady surface waves on a fluid of infinite depth to the solution of an integral equation.

Later he extended his analysis to waves on a fluid of finite depth. The

analysis.i.s, however, complicated and the final wave profile is not given explicitly by the: integral equation.

In 1964 Mime-Thomson solved the integral equation numerically and obtained the wave profile for the case of the solitary wave. He

also summarizes the theory of this integral equation in the fifth edition of his book (Mime-Thomson, 1968).

In this paper we will extend the method used by Long to, .obtain a new exact equation for steady surface waves which, unlike the equation derived by Nekrasov, will apply to all waves including solitary waves and waves of maximum height.

2. Derivation of the Exact Integral Equations

We assume that the fluid is incompressible and inviscid and that it flows over a horizontal bottom. The motion is assumed to be

two-dimensional irrotational and steady. The coordinate axes are chosen so that the bottom is y = 0 and the waves are stationary. This system of coordinate axes is shown in Figure (1).

Bernoulli's equation states that

p/p + 1/2 q2 + gy = constant (2.1)

on each streamline and in particular on the free surface, where the pressure is constant,

+

g(y5 -

where the subscript s denotes a value on the free surface and C1 is

Since it has been assumed that the fluid is inviscid and that the motion is two dimensional and irrotational, there exists a velocity

potential 4 anda stream function i such that + i is an

analytic function of x + i y and vice versa. Therefore

q = (,p)/(x,y) = 1/ -

l/{((ax/)2

(2.3)

Combining this identity with the previous equation, we obtain

(x/a)

(2.4)

The equation for y(,ip) is

Since

0 , (2.5)

with the boundary condition

y=0 on p=O

If we denote the Fourier transform of y with respect to c by bars then

(k,p)

(2Tr)_l'2,

_f

(2.6)

A(k)

=2y/p

implies that

ax/c = k A(k) cosh(k) _(2.9)

Therefore

= k tanh(kP ) (2.10)

After noting that k tanh(kp) is the Fourier transform of -T(' log tanh

( rr /ip) we can invert equation (2.10) to obtain

y= -

/p)

In particular along the free surface

= we have

1/2

- ir

f

(y5-h)} -

where [F ()] denotes

F(-).

(2.12)

It will be useful to write this in non-dimensional form. This we can do by introducing C1 as a velocity scale and h as a length scale. Using

capitals for non-dimensional quantities we have

=

-[

y)] lo

wII/)

where

F=c1/VW

We can now define y(,ip) over the Whole flow field because we know

y on = 0 and i

= for all

Thus

where is the solution of

equation

3. Derivation of Existing Theories From The Integral Equation

By using the appropriate approximation we can derive existing theories for steady surface waves.

Stokes Waves

The Stokes wave is derived by assuming that the equation for the height

above the height of the trough can be linearized in n

From equation (2.12) we have

1

y5(-c1)

dc0

y(cp) =

Sfl

h +

= -

I

-2-

(1 +

--

(c-)

1 1

(2.14)

IJ/p5)

The solution of this linear integral equation that is zero at the trough is

= a( 1 + sin(kc)) 3.2)

where a and k satisfy the equations

h+ a

( 1+ ga/c)

The equation that determines x in terms of ct is, from

equation (3.4),

S}

2}112

Cl ={l/{l/ {l C1

_Ill

1

which, to the first order in , gives

x=-! ((l +!_)

cosh)

3 k

1 1

(3.5)

(3.6)

If we introduce a wave length A and a nondimensional potential x by

k=2rr/Ac1 ,=k

then the solution becomes

ri = a( 1 + sin x) (3.8)

x =

s-

2ir

This expression for the height given parametrically by equations (3.8) and (3.9) is a trochoid and not a sine wave, as in the Stokes solution. However, the first term in the expansion of the independent variable, x , is of zeroth order in a . The term of order a has a

second order correction to theperturbation r of the free surface and

should thus be neglected for a consistent linear solution. If it is neg-lected then the solution becomes the same as the Stokes solution

= -

P() +

,1 ()

Applying this result to the right hand side of equation (3.11) _we

obta in

1 +N()

+4+-

Therefore

N" 2 (l'P) +

(F2

'P2

The solution of Equation (3.14) that is _{zero at the trough is}

N = a

where a, , and k are given by the solution of

2 2 'Ps

N"

+---.z.

log tanh ( -

kI/'P5)

(3.12)

8 Cnoidal Waves

We can now recover the second order differential equation _for

cnoidal waves by making the accumption that N , the nondimensional height

above the trough, is of order a and that d2M

N/d2M

(a1)

retaining the terms up to order a2

Thus equation (2.13) becomes

1 + N = - [1 + N/F2 + _{-N2/F2] log} tanh

_/4icI'Y

(3.11)

We now use the result that if P is a slowly varying function

-r

_f

P(-)

ki /'i'5

(2k2 - 1) a'P = k2 F2(F2

2 3

aF2

k21Y2

S

'

,

From equations (3.16) and (3.17), we can see that if this solution is

to be consistent with our assumption that a is small then (F2 - 1) and

(l _{- '')} are both of the order a and

a

4

+8(1

k2

= .+

-1/2

(3.20)

approximately where we have neglected terms of order (F2 - 1)2 and

(1

-Since 0 < k < 1 then we must have < 1

The differential equation that determines X is obtained as

dX = F2/(F2 2N) -Therefore X = {1 + 2a2 C

-

Fk

Fk

where K and E are the complete elliptic integral of the first and

second kinds respectively, and Z(a , k) is the Jacobi Zeta function.

As in the linear theory this differs from the ordinary first approxi-mation to cnoidal wave only in the higher order terms in the expansion of the independent variable. These should be neglected 'to give the first

10

approximation as

N = a Cn2

(a X, k)

We can proceed to the second and higher approximations by looking for a solution of the form N = N1 + N2 + . . where N = 0 (a1)

and expanding the integral in equation (2.13) to obtain the terms of the desired order. This isan extension of the method used by Long (1955)

who obtained a higher order approximation to the solitary wave. Surface Waves of Maximum Amplitude

In 1880 Stokes conjectured that surface waves of maximum height would have a sharp peak with an enclosed angle of 120°. This would correspond to a relative local velocity of zero at the crest. Although no proof has been given, all estimates of the limiting height that have been referred to in this paper., apart from the estimates of Laitone, have used this criterion. We cannot prove the existence of a solution with a 120° wave crest, but we will show (see Appendix) that, any solution of equation (2.13) whether

periodic or of the solitary wave type always has a corner when the. ampli-tude is 1/2 'F2.

Numerical Solution for Large Amplitude Solitary Waves

A.numerical solution to equation (2.13) was attempted for the case of the solitary wave, that. = 1 . Solutions were sought for the range of

values of the Froude Number that gave large amplitudes. By largeampli-tudes we mean amplilargeampli-tudes that are less than the maximum (.8), but are too great for 'shallow water theory to apply.

The point c _{= 0.} was, taken as the crest so that the profile was an

even function of . Denoting equation (2.13) by

where

E(cI) = 1 + N

+

:

F2-2N

N'2)1"2] log tanh ( kI) d

(5.2)

and N = Y5-l.

We define residuals ...2M)by

= _E(JH) , (5.3)

where H is a suitably chosen step length.

The solution N=NCiH)

j =

> 2M the solution was taken to be N exp (2MH - ) where is

the smallest root of F2 I3 tans (for a proof that N exp- k1

as see appendix)

The equations

.LN

+ = 0,

(5,

were solved for corrections so that a better solution N + EN.

was obtained. This process was repeated until the ziN were small.

The equation

.2

(5.5)

S

F-2N

was then integrated to obtain the physical coordinate X.

6. Discussion and Interpretation of the Results

For values of F less than about 1.2 the successive approximations converged very quickly to the solution. As F increased beyond 1.2 the number of iterations needed increased rapidly until it was necessary to take very small steps in F and predict a very accurate first guess from

12

possible to obtain solutions for values of F up to 1.29. The reason

for this slow convergence is that as F increases 2/F2 times the amplitude increases. This meant we were approaching the singularity in

12

the integrand which is present when the amplitude is equal to F

It was not possible to obtain a solution for this critical value of F

by this process, but by plotting F against the amplitude an estimate of the critical value of F was obtained.

The numerical results are shown in Figures (1) and (2). It is

interesting to compare them with the known first and second approximations obtained by Laitone (1960). This will serve as a useful guide to the

convergence of the perturbation expansion for the solitary wave. The

wave velocity has been worked out to the fifth order by Long (1955) and the higher order approximations become successively better for all values of the amplitude (Figure (3)). However Figures (1) and (2) show that the first approximation is in fact better than the second for the range of values of F used. For example, the curvature at the crest in the second approximation is less than the curvature in the first ap-proximation, whereas the exact solution has a greater curvature than the first approximation. It has been observed experimentally (Stephan and Dailey (1953) that the crest of a solitary wave is sharper than that of the first approximation. Again at infinity the first

approxi-mation is nearer the exact solution than is the second. These results

are rather strange, because the first approximation is such a good one and the coefficient of the second order correction is small compared

We can explain this by looking at the form of the solution as

x -' . The solution of the exact equation tends to _{zero like}

exp _(-IxI) as x + where is the smallest root of _tans

In the perturbation solution of Laitone, the behavior _{at infinity is}

given by exp

_(-2cIxI),

If we now rearrange this using the series for F _{(Laitone 1960 or Long} 1955) given by

F2 = 1 + a a2 _(6.2)

we obtain

CL - i{3(F2-1)}(l - 3/5(F2-l) + O(F2-l)512

We can now obtain further terms by expanding F2 = tan 2CL in a

power series in F2-1.

Thus CL = .- v'{3(F2-1)} ( 1-3/5 (F2-l) + 394/700 (F2-l)2 - 948/1750(F2-1)2

+

(6.3)

This series converges slowly and a large number of terms are required

to give an accurate value of x for large values of F2-1 . The

second approximation for c is smaller than its true value. Thus the

horizontal length scale is increased and the wave profile is given a

linear stretching in the horizontal direction. The error in neglecting

the higher order terms in the expansion for a is magnified at large

values of x

14

7. Concluding Remarks

We have found an exact integral equation for the height of a sur-face wave and shown that it allows solutions with a 1200 corner under certain circumstances. The numerical solution given in the case of the solitary wave shows that as the Froude number increases the solution tends to this special solution with a corner. These solutions have shown the'limitations of. the solutiOn obtained by the perturbation expansion method. Although the technique used to solve the integral equation was not applicable for the special solution, we could estimate the amplitude and Froude number of the maximum height solitary wave as

amax = .86 ,

Fa = 1.31

There figures were obtained by curve fitting through the amplitude Froude number relations obtained by numerical analysis.

The results are about 4% higher than most estimates but this could easily be accounted for by the inaccuracy of the numerical integration and extrapolation. For an accurate estimate we would need a different numerical method than the one used here, so that a solution for the maximum solitary wave could be obtained.

APPENDIX

In this appendix we prove two results used in this paper That any solution of equation (2.13,) has a corner of 1200

when the amplitude is 1/2 F2

That all non-periodic solutions of equation (13) tend to

zero as like

exp(-II).

F2 tan.

1) Equation (13) may be written as

I

- )

)I

(A.1)

where then , F2 dN 2 . S

-

F-2N

Y=l+N

The only singularity in the surface occurs when N = - F2 . If in

this casethe. form near the crest is given by

( .

AI,I_a/2

(ct/2)

as + 0

or X.= 2A/(2-) sign ()

iI1_2

(A.5)

We will now show that this form satisfies equation (A.l)in the neighborhood of the crest. By substituting equation (A.3) with equation (A.1), we obtain

l+F2_BI1a +

C ) log tanh (

cz+/y5)

y) d.

(A.6)

l

[AI!'

I-I/)

16

Equating the two lowest powers of gives

a = 2/3 B =

/Y

tanh d

If we neglect all O(), equation (5) can be reduced to

+ .-

(I2-2I/2)dc

(A.7)

The integral on the right hand side can be evaluated by making the substitution

2/2cs

fa/2log

_(I2-c2l/2)

L 1-cx/2

So that equation (A.6) becomes

2

B = cOnst - 2A 1(2-a) 2 cot .

So that the form near the crest becomes

provided < Tr

N-1F2

2

that is a corner of 1200

2) _{For thecase of a non periodic wave, we have by definition}

= 1 . Thus equation (13) can be written as

.2

N(c)

0

F-2N

- 1] log tanh -

I-?;Itanh

Id?;

(A.13)

If we now look for a solution of the form

N(.) = A

eI1

(A.l4)

N() = bounded for _{< !- I}

then the integral on the right hand side can be split into three integrals,

I. over the range to

, 12 from 1/2 to and 13 from 0 to 1/2

By a suitable change of variable, we can show that

00 f e log tanh - d irE 0 +

(e11)

_Q

A.eY.w

je

(e1)

o(e'')

18

Thus substituting for N() I, ₁₂ and 13 in equation (A.13) yields

Ae =

- wF2

e'

f

(et

e1) log-tanh

Thus equating the coefficient of gives tan

RE FERENCES

Boussinesq, J. 1877 Essai sur la theorie des eaux courants. Mem.

Pres. Acad. Sci. Paris, 23.

F.riedrichs, K. 0. 1948 J. Comm. P. and A. Math. 1, 81. Keller, J. B. 1948 J. Comm. P. and A. Math. 1, 323.

Korteweg, D. J. and De Vries, G 1895 Phil. Mag. (5) 39, 422. Laitone, E. V. 1960 J. Fluid Mech. 9, 430.

Lenau, C. 1966 J. Fluid Mech. 26, 309. Long, R. R. 1956 Tellus, 8, No. 4, 1460.

McCowan, J. 1894 Phil. Mag.(5) 38, 351.

Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. MacMillan,

Packham, B. A. 1952 Proc. Roy. Soc. A, 213, 238. Lord Rayleigh 1876 Phil. Mag. (5) 1, 247.

Russell, S. 1837 Rep. Brit. Ass. p. 417. Russell, S. 1844 Rep. Brit. Ass. p. 311.

Stephan, S. C. and Dailey, J. W. 1953 _{T.A.S.C.E. 118, 575.} Yamada, H. 1957 Rep. Res. Inst. App. M. Kyushu Univ. 5, 18, 53.

Fifth Edition.

Nekrasov, A. I. 1921 _{Izv. Ivanovo-Voznesensk Politekhn. Inst. 3, 52.}

Nekrasov, A. I.

or

1922

1951

Izv. Ivanovo-Voznesensk Politekhn. Inst. 6, Moscow: _{Izdatel'stvo Akad. Nauk. S.S.S.R.}

SECOND

A PPROX..

SECOND

APPROX.

SECOND

A PPROX.

FIRST

APPROX.

FIRST

APPROX.

FIRST

APPROX.

EXACT

SOLUTION

EXACT

SOLUTION

--0.4

0.2

-

I-EXACT

-O.'+

SOLUTION_

w

-0.2w

_>

--0.6

0.4

0.2

-I

0

HORIZONTAL DISTANCE

FIG. 2

AMPLITUDE O.492

AMPLITUDE =

0.593

AMPLITUDE 0.686

2

-I

0

HORIZONTAL DISTANCE

FIG.

2

5ION

PLITUDE =0.716

FIRST

APPROX.

-0 2

I

0752

0.2

li

w

I6

0

(I)

0

0::

LL.

1.0

0

3

4

56

7

AMPLITUDE

FIG. 4

8

MAXIMUM

AMPLITUDE:O.86

II.

12

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3. REPORT TITLE

An Exact Integral Equation for Steady Surface Waves

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Technical Report

5. AUTHOR(S)(Last nanie. Ii ret name. initial)

John Byatt-Smith

6. REPORT DATE

February 1969

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13. ABSTRACT

An exact integral equation is found for inviscid steady surface waves.

Existing known approximations are all derived from the one equation. A

numerical solution is attempted in the case of the solitary wave. From the

numerical solution the amplitude of the solitary wave of maximum height is estimated. _{The numerical solution is also compared with the existing}

approximations and the implications drawn from the comparison.are discussed.